M. Vysotskyy and V. Vysotskii National Taras Shevchenko University of Kyiv

1. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals M. Vysotskyy and V. Vysotskii National Taras Shevchenko University of Kyiv

2. In the report the methods of optimization of nuclear processes at interaction of low energy proton or deuteron beams with oriented crystal targets are considered. These methods are connected with the peculiarities of coherent correlation effects at oriental motion and channeling of charged particles in crystals. Similar processes of optimization of controlled fusion in crystals (but without the use of coherent correlated states) were examined for 7-8 years before historical experiments of Fleischmann and Pons. [Vysotskii V.I., Kuzmin R.N. Soviet Phys. -J.T.P. Letters), v.7, #16, 1981, p. 981-985; Vysotskii V.I., Kuzmin R.N. Soviet Phys. -J.T.P.), v. 53, № 9, 1983, p. 1861-1863] In last time a lot of experiments on investigation of low-energy beam-target interaction were conducted (e.g. [A. Kitamura, Y.Awa, T.Minary, M.Kubota, A.Tamiika, Y.Furuyama. Proc. 10th ICCF, p. 623-634; A.Huke, K.Szerski, p.Heide, Proc. 11th ICCF, p.194-209]). Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

3. Channeling effect If the direction of a charged particle incident upon the surface of a monocrystal lies close to a major crystal direction, the particle with high probability will suffer a small-angle scattering as it passes through the several layers of atoms in the crystal. If the direction of the particle's momentum is close to the crystalling plane, but it is not close to major crystalling axes, this phenomenon is called "plane channelling". Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

4. Avaraged potential of crystal plane Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

5. Positive particles (positrons, protons) Negatively-charged particles like antiprotons and electrons are attracted towards the positively-charged nuclei of the plane, and after passing the center of the plane, they will be attracted again, so negatively-charged particles tend to follow the direction of one crystalline plane. Positively-charged particles like protons and positrons are repulsed from the nuclei of the plane, so they tend to follow the direction between two neighboring crystalline planes, at the largest possible distance from each of them. The positively-charged particles have a smaller probability of interacting with the nuclei and electrons of the planes. Negative particles (electrons) Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

6. Is it possible to increase the possibility of proton interaction with crystal nuclei? (for example for the optimization of the nuclear reactions using the scheme of nuclear synthesis based on a beam of accelerated particles, incident on the LiD or LiT crystals). Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

7. Studies of particle channeling process in a crystal are based on a supposition concerningthe mutual independence of particle quantum states ateach level of transverse motion. This supposition isincorrect for the initial region ofchannels and for the total length of nanochannels, to which hollow channels in zeolites, asbestos filaments and, to a lesser extent, in carbon nanotubes and fullerenes correspond. In these areas the wave function of a particle is formed during the process of the joining of its states prior to entry into a channel and the superposition of possible states in the crystal channel, and corresponds to mutually coherent states. The special nature of these states is related to the possibility for the manifestation of a different type of interference effects. The most interesting among these is the process of coherent correlated state formation Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

8. The quantitative characteristics of this correlation are determined by the correlation coefficient r The unlimited growth of dispersions at r → 1 leadsto the possibility of much more efficient penetrationof the particle with a small transverse motion energyin the region below the barrier than for the sameparticle in the uncorrelated state (at r = 0). Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

9. How coherent correlated state can be formed? The possible method for the formation of the correlated statewith|r| → 1 is the deformation of the potential well structure The method for formation of thecorrelated state with |r| → 1 which doesnot require considerable deformation of the potentialwell parameters is needed. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

10. The simplest method for excitation of the particle correlated state is related to nonstationary deformation of the harmonic potential V(x) in the field of which this particle is situated, i.e., virtually to the deformation of the harmonic oscillator. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

11. The equation for theharmonic oscillator and the corresponding initial conditions in the dimensionless form Let us present the solution as a complexfunction: In this case the final expression for r The system of equations makes it possible to find the exponent for the amplitude of oscillatorfluctuations according to the given lawfor the change in ω(t), and to find r(t) onthe basis of obtained expression α(t). Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

12. It is possible to find r(t) forany law of the change in ω(t), independent of whatcauses this change, either due to the change in thepotential well parameters V(t) and α(t) or even due to thechange in the particle mass M(t). We can determine the nonstationary dynamics of correlated state formation underdifferent modes of change in the frequency ω(t). Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

13. |r(t)| 1.0 0.9 d) 1 2 3 4 5 67 8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0t 0.0 80 70 60 10 20 30 90 50 40 0 1. Thecase of a monotonic change of channel parameters A monotonic decrease in the harmonicoscillator frequency ω(t) leads to an increase in α(t),which leads to an increase in the amplitude of its oscillations and the correlation coefficientup to its limiting value |r| → 1 Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

14. 1. Thecase of a monotonic change of channel parameters The impracticability of this method for formationof the coherent correlated state, for example, duringparticle motion in theinterplane crystal channel isobvious. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

15. 2. Thecase of CCS formation at the harmonic law of channel parameters change limited range Such a mode can be achieved, e.g., for the constantpotential well depth V in which the particle is situated and the periodic change in its width a in the range Or for the constant well width a andsmall change in its depth in the range Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

16. 1.0 1.0 1.0 |r| |r| c) |r| b) 0.8 0.8 0.8 a) 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 40 w0t w0t w0t 80 80 20 40 20 40 60 60 0 0 160 80 120 0 2. Thecase of CCS formation at the harmonic law of channel parameters change Time dependence of the correlation coefficientr(t) for the case of the limited periodic changein the oscillator frequency ω(t) = ω0(1 + gΩcosΩt) at Ω = ω0 for differentvalues of the frequency modulation index: (a)gΩ = 0.1; (b) gΩ = 0.2; (c) gΩ = 0.3. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

17. 2. Thecase of CCS formation at the harmonic law of channel parameters change It is seenthat when g increases, the rate of change in the correlation coefficient with time increases and its valueapproaches that of the limiting |r| → 1 faster, providing complete translucence of any potential barrier. An increase in the modulation frequency from Ω = ωto Ω = 2ω leads to sharp growth in the rate of increasein the correlation coefficient’s (|r(t)|) maximum value,which approaches the limiting value very fast even at asmall g = 0.01 value Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

18. 1.0 1.0 1.0 1.0 1.0 |r| |r| |r| a) c) |r| 1.0 d) b) a) b) |r| |r| 0.8 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 w0t w0t w0t 0.0 w0t 160 160 40 80 80 120 40 120 0 0 160 160 40 80 40 80 120 120 0 0 w0t w0t 40 38.5 39 39.5 40.5 38 40.05 40.1 40.15 Time dependence of the correlation coefficient r(t) for a change in the oscillator frequency atΩ = 2ω for different values of the frequency modulation index: (a)g = 0.01; (b) g = 0.025; (c)g = 0.05; (d)g = 0.1. Thefragment marked in fig. d is shown in a larger form in below Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

19. 2. Thecase of CCS formation at the harmonic law of channel parameters change The structure of the expression |r(t)| is characterized by a fast increase in the amplitude |r(t)|to the maximum value |r| → 1.Very narrow hollowsare observed in the gaps between the extended maximafor |r| ≥ 0.99, caused by the interference phenomena. The width of these hollows sharply decreaseswhen time increases. It is seen from these results that the relative width ofthese hollows is extremely small, not exceeding 1–2%at t ≈ 40/ω, and decreases to zero for further increasesin t. The remaining part of thedependence |r(t)| corresponds to the almost constant and maximum possiblevalue |r| → 1. Let us consider the possibility for implementing this effect during channeling. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

20. The nonstationary modulation of the well parameters corresponds to a periodicspatial modulation of the well parameters. Taking into account that the velocity of themotion of particles in the channel is always muchhigher than the velocity of an ultrasound (US) wave,this modulation can be achievedduringthe excitation of the longitudinaltraveling US wave inthe crystal along the channel direction, the effect ofwhich leads to periodic changes in the longitudinaldistance a between the atoms, and also the height ofthe channel wall. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

21. The analysis performed above showsthat the relation between Ωand ωclose to optimal is Ω = 2ω, which corresponds to thenecessary spatial modulation period of the channel walls (equal to the wave length of the longitudinal US oscillations) Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

22. Using a typical crystal with Vmax(0) = 20 eV a0 = 1.5 × 10–8 cm Proton beam with the energy E = 100 keV v = 4.5 × 108 cm/s The length of the US wave necessary for forming the correlated state is Λ ≈ 0.4 μm. The length of the US wave isthe same for adeuteron beam with the same energy. This wave should have an amplitude that ensuresthe necessary maximum relative change in the interatomic distance Δa/a = g = 0.1–0.01, leading to ananalogous dynamic change in the potential barrierheight. All necessary parameters determined for thisexample, are quite real. Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals

23. When these conditions are met, the correlationcoefficient for a particle (proton or deuteron),movingin the channel, even at a small value of the modulationindex g, increases very quickly. Since for these particles the correlation coefficient can reach values closeto|r| = 1, one should expect manifestations of theanomalous character of their interactions with crystalnuclei, including optimization of the nuclear reactions. In particular using a deuteron beam, asharp increase in the efficiency of nuclear synthesisshould be expected when it passes through such amodulated crystal, containing deuterium or tritiumnuclei (for example, when using the scheme of nuclearsynthesis based on a beam of accelerated particles,incident on the LiD or LiT crystals). Subbarier Nuclear Interaction of Channeling Particles at Self-Similar Formation of Correlated States in Periodically Strained Crystals